The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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64 3. <strong>The</strong> derivation of nuclear forces from chiral Lagrangians<br />
form we should require the same transformation property (3.104) under go E SU(2)v X SU(2)A'<br />
Us<strong>in</strong>g eqs. (3.98), (3.100) and (3.101) one verifies that uJ.L <strong>in</strong>deed satisfies this condition, i.e. that<br />
and thus corresponds to the matrix representation of the covariant derivative of �.<br />
(3.105)<br />
<strong>The</strong> pion fields can be def<strong>in</strong>ed by a particular parametrization of U. Choos<strong>in</strong>g aga<strong>in</strong> the 2 x 2<br />
matrix representation (3.78) of the group generators Ai, Vi we can parametrize U, for example,<br />
as follows:<br />
Z'Fr • T<br />
U=exP ( h ) , (3.106)<br />
where 7ri are pseudoscalar fields (pions) and f1f is some constantP This also uniquely def<strong>in</strong>es the<br />
quantity uJ.L <strong>in</strong> terms of the pion fields. It is easy to construct chiral <strong>in</strong>variant terms consist<strong>in</strong>g of<br />
u/s. One simply has to build traces ( ... ) <strong>in</strong> flavor space of any product of the uJ.L's. For <strong>in</strong>stance,<br />
the term (uJ.Lu J.L ) is clearly chiral <strong>in</strong>variant:<br />
(3.107)<br />
Let us now <strong>in</strong>troduce the nucleon field \]J. We will require that the nucleon field belongs to the<br />
CCWZ realization and transforms under go E G via eq. (3.54):<br />
(3.108)<br />
S<strong>in</strong>ce we have chosen the representation (3.78) of the group generators, h(go, 'Fr) is a 2 x 2 matrix<br />
<strong>in</strong> the flavor space. Note that h(go, 'Fr) becomes <strong>in</strong>dependent on 'Fr for all go E SU(2)v and forms<br />
the fundamental representation of SU(2)v. To def<strong>in</strong>e the covariant derivatives of the nucleon field<br />
it is convenient to <strong>in</strong>troduce the so-called connection r J.L:<br />
It follows from eq. (3.98) that r J.L transforms as:<br />
Here we made use of the similar trick as <strong>in</strong> eq. (3.103):<br />
<strong>The</strong>refore, the covariant derivative of the nucleon def<strong>in</strong>ed as<br />
transforms <strong>in</strong> the same way as the field \]J (covariantly):<br />
(3.109)<br />
(3.110)<br />
(3.111)<br />
(3.112)<br />
(3.113)<br />
Thus, any flavor scalar built up from isosp<strong>in</strong>ors \]J, D J.L \]J with <strong>in</strong>sertions of uJ.L is chiral <strong>in</strong>variant.<br />
17 It can be shown that <strong>in</strong> the chiral limit f" is the pion decay constant.